The XIXth century was definitely a time for extensive culinary experimentation in France, as a growing share of the population could afford meat and more complex meals. Dog meat was eaten in the immediate aftermath of the 1870 French-Prussian war in Paris (with the siege and the Commune de Paris episode) due to scarcity^1, and some boucherie canines are attested (here is a gravure from the 1870s-1880s). An example would be this one which claims to be the first in Paris (in 1910). This is probably inaccurate as some dog meat was definitely eaten as early as 1870 but it may be the first one in an organized supply chain with dogs slaughtered in relatively correct hygiene conditions. It was a relatively common meat for the poorest share of the population in Paris between 1900 and 1920, with traces in the literature of the time.
An example would be the passage in the poem series Alcools, by G. Apollinaire (who basically ushered French poetry into modernity : we could call him the Stravinsky of French poetry) with the verses in La maison des morts :
« Quelques-uns nous quittèrent Devant une boucherie canine Pour y acheter leur repas du soir »
Some left us In front of a canine butchery To buy their supper
If I remember correctly there is also a reference in L.-F. Céline (also describing the Paris population of the early 1900s) in Mort à Crédit (will look for it).
To answer the why question, people had far less taboos at the time and would already eat offalls and parts of an animal a modern american wouldn't probably try (though soul food has more freedom of experimentation) -- French cuisine still prepares various delicacies from non-muscle parts (sweetbreads with cognac for the fanciest, tripes à la mode de Caen and other pieds de cochon for the least fancy). It was a time where the population would try to have meat relatively frequently and where dog would be an affordable meat for the working class in Paris. Very similarly, horse meat was authorized for human consumption in 1866^2: while horse was (for religious reasons and as a useful animal considered noble) eaten only in extreme circumstances until the XVIIIth century, its consumption was famously practiced by the remains of the Grande Armée on the way back from Russia. In the late XIX century, it became an inexpensive source of meat with parts similar to beef (you can find rumpsteak or a roast in a horse) and good nutritional qualities. Horse consumption was actually defended by the first animal rights groups in the 1850s, to avoid that horses would be used to their exhaustion and give them a more "merciful" death in slaughterhouses^3. It was in 1900s mostly a middle class meat, as the rural population would appear not to be keen on eating their working mates^4. To this day, horse consumption by humans represents around 15000 horses a year in France, with 250 specialized butcheries in the country (around 300g/inhab./year), with an esteemed share of the population of 18% consuming it. It is actually quite good and similar to beef, only leaner and with a sweeter afertaste.
^1 : X. Malher et B. Denis, Le chien, animal de boucherie, Ethnozootechnie, Société d'ethnozootechnie, no 43, 1989, p. 81-84
^2 : Sylvain Leteux. L’hippophagie en France : la difficile acceptation d’une viande honteuse. Terrains et Travaux : Revue de Sciences Sociales, ENS Cachan, 2005, pp.143-158.
^3 : Kari Weil, They Eat Horses, Don't They ? Hippophagy and Frenchness, Gastronomica, vol. 7, no 2, 2007
^4 : Éric Baratay, Et l'homme créa l'animal: histoire d'une condition, Odile Jacob, coll. « Sciences humaines », 2003
